In the field of geometrical layout for railroad tracks the traditional elements have been the straight line (with constant curvature equal to zero), the circular arc (with curvature that is constant but not zero), and the spiral (along whose length the curvature varies monotonically). When two sections of track that have different constant values of curvature would otherwise meet one another it is normal (with exceptions in some special cases) for the two sections to be connected by a spiral whose curvature and compass bearing at each end matches those of the adjacent section to which that end connects. Spirals have traditionally been conceived as geometrical shapes on the ground, and a number of specific shapes have been devised and applied during the past two centuries.
A method for the design of railroad track spirals, and a number of specific shapes that can be obtained using this method, are described in International Application No. PCT/US01/41074 by Louis T. Klauder, Jr., titled “Railroad Curve Transition Spiral Design Method Based on Control of Vehicle Banking Motion” (hereafter referred to as the “KS_Method”). The KS_Method looks at a spiral not first of all as a shape but rather as means of helping the guided vehicles change their roll or bank angle from the value appropriate for getting gravity to provide centripetal acceleration in one section to the value appropriate for that purpose in a following section whose centripetal acceleration is different.
The following introduces some terminology that is helpful for describing the general field of track and guideway geometry, the KS_Method, and the methods of the current invention. Let the speed of travel be denoted as v, and denote compass bearing of the track as a function of distance s along the track by b(s). The curvature is defined as the first derivative of the bearing with respect to distance, which is denoted as db(s)/ds. The component of centripetal acceleration in the plane of the track will be provided by gravity if the equationv2db(s)/ds cos(r(s))=g sin(r(s))  (1)is satisfied, where g is the acceleration of gravity, r(s) is the function that specifies the roll or bank angle of the track as a function of distance s, and cos and sin are the common trigonometry functions. Hereafter, the forgoing equation is referred to as the “balance equation”, and v is referred to as the “balance speed”.
In the KS_Method for designing a spiral the first task is to choose a functional form for r(s) within the length of the spiral. The subsequent tasks are: to integrate the balance equation to obtain the compass bearing b(s), to integrate cos(b(s)) and sin(b(s)) to obtain respectively the x and y coordinates of points along the spiral, and to identify the parameters of the function r(s) for which the resulting shape properly connects to the adjacent sections of constant curvature track, incorporating the forgoing two stages of integration into an iterative search for that purpose if need be. A transition shape connects properly to an adjacent straight or circular arc section if the end of the shape has a point in common with the line or arc and if the shape has the same compass bearing and curvature as the line or arc at the point in common. The most prominent parameter of r(s) is normally the length of the spiral.
Approximations can be introduced to simplify equation (1) (the balance equation) and the integrals of cos(b(s)) and sin(b(s)) to obtain x and y respectively. The most common simplification replaces each cosine function by unity and each sine function by its argument expressed in radians. These simplifications will hereafter be referred to collectively as the “small angle” approximation. If the roll function is a polynomial in s and this simplification is applied, then both stages of integration called for in the KS_Method (and in the method of the current invention) can often be done in closed form so that numerical integration and iteration are not required. This simplification provides a good approximation to the extent that r(s) and b(s) are both ≦0.1 throughout the transition. Even when these two angles do not stay that small, this approximation, while not so good mathematically, may still give geometries that are effective in practice.
The method of the present invention takes advantage of the previously known principle that the axis about which the roll of the track takes place does not need to be located in the plane of the track but can be at a specified height, which height is also a parameter of the spiral.
The method of the present invention provides solutions for two existing problems in the field of railroad track transition curve geometry. One problem can arise when an existing route is being upgraded to allow operation at higher speed. If for a particular curve the speed increase is being provided for by increasing the superelevation (or banking) and without change of the radius of or path followed by the curve, then the offset between the curve and a neighboring straight section will be unchanged and the length of a standard spiral connecting them will be unchanged. The offset is the shortest distance from a circular extension of the curve to a straight extension of the straight section. It is generally necessary in such a case to find some way to lengthen the spiral. Examples of ways that traditional spirals and circular arcs have been used to address this problem in the past can be found in the article titled “Optimation of transition length increase” by Henryk Baluch, published in the October 1982 issue of Rail International.
The other problem occurs when maintenance work is being planned to adjust the alignment of an existing spiral whose shape has become deformed by passing trains. The problem is whether, and if so how, to mathematically define the shape to which the spiral should be restored. If a system is in place for measuring the location of the track relative to local fixed monuments and the original shape was mathematically defined and the existing shape has not drifted very far from the original shape, the spiral can be restored to the original design shape. When the forgoing conditions are not all met, the practice has normally been to “smooth” the alignment so that curvature measured along the corrected alignment becomes close to some form of running average of the curvature of the previous deformed alignment. Alignments created by smoothing of that kind have generally not been described by mathematical formulae. As a result, alignments have tended to drift over time.